What is the integral of xsqrt(x)sec^2(sqrt(x^5)) ?

The integral of xsqrt(x)sec^2(sqrt(x^5)) is 2/5 tan(x^{5/2})+C

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integral of xsqrt(x)sec^2(sqrt(x^5))

\int x x sec^{2} (x^{5}) d x

\frac{2}{5} tan (x^{\frac{5}{2}}) + C

Solution steps

\int x x sec^{2} (x^{5}) d x

Simplify

x x^{\frac{1}{2}} : x^{\frac{3}{2}}

= \int x^{\frac{3}{2}} sec^{2} (x^{5}) d x

Apply u-substitution

: \int \frac{2 sec ^{2} ( u )}{5} d u

= \int \frac{2 sec ^{2} ( u )}{5} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{2}{5} \cdot \int sec^{2} (u) d u

Use the common integral:

\int sec^{2} (u) d u = tan (u)

= \frac{2}{5} tan (u)

Substitute back

u = x^{5}

= \frac{2}{5} tan (x^{5})

Apply radical rule:

n a^{m} = a^{\frac{m}{n}},

assuming

a \geq 0

= \frac{2}{5} tan (x^{\frac{5}{2}})

Add a constant to the solution

= \frac{2}{5} tan (x^{\frac{5}{2}}) + C

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What is the integral of xsqrt(x)sec^2(sqrt(x^5)) ?
The integral of xsqrt(x)sec^2(sqrt(x^5)) is 2/5 tan(x^{5/2})+C